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Properties

Label 2-3380-3380.327-c0-0-0
Degree $2$
Conductor $3380$
Sign $0.532 - 0.846i$
Analytic cond. $1.68683$
Root an. cond. $1.29878$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.278 − 0.960i)2-s + (−0.845 + 0.534i)4-s + (−0.160 − 0.987i)5-s + (0.748 + 0.663i)8-s + (−0.721 − 0.692i)9-s + (−0.903 + 0.428i)10-s + (−0.316 + 0.948i)13-s + (0.428 − 0.903i)16-s + (−0.871 + 1.74i)17-s + (−0.464 + 0.885i)18-s + (0.663 + 0.748i)20-s + (−0.948 + 0.316i)25-s + (0.999 + 0.0402i)26-s + (−1.53 + 0.444i)29-s + (−0.987 − 0.160i)32-s + ⋯
L(s)  = 1  + (−0.278 − 0.960i)2-s + (−0.845 + 0.534i)4-s + (−0.160 − 0.987i)5-s + (0.748 + 0.663i)8-s + (−0.721 − 0.692i)9-s + (−0.903 + 0.428i)10-s + (−0.316 + 0.948i)13-s + (0.428 − 0.903i)16-s + (−0.871 + 1.74i)17-s + (−0.464 + 0.885i)18-s + (0.663 + 0.748i)20-s + (−0.948 + 0.316i)25-s + (0.999 + 0.0402i)26-s + (−1.53 + 0.444i)29-s + (−0.987 − 0.160i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.532 - 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.532 - 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3380\)    =    \(2^{2} \cdot 5 \cdot 13^{2}\)
Sign: $0.532 - 0.846i$
Analytic conductor: \(1.68683\)
Root analytic conductor: \(1.29878\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3380} (327, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3380,\ (\ :0),\ 0.532 - 0.846i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2757232503\)
\(L(\frac12)\) \(\approx\) \(0.2757232503\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.278 + 0.960i)T \)
5 \( 1 + (0.160 + 0.987i)T \)
13 \( 1 + (0.316 - 0.948i)T \)
good3 \( 1 + (0.721 + 0.692i)T^{2} \)
7 \( 1 + (0.799 - 0.600i)T^{2} \)
11 \( 1 + (0.999 + 0.0402i)T^{2} \)
17 \( 1 + (0.871 - 1.74i)T + (-0.600 - 0.799i)T^{2} \)
19 \( 1 + (-0.866 + 0.5i)T^{2} \)
23 \( 1 + (0.866 + 0.5i)T^{2} \)
29 \( 1 + (1.53 - 0.444i)T + (0.845 - 0.534i)T^{2} \)
31 \( 1 + (0.663 - 0.748i)T^{2} \)
37 \( 1 + (0.0258 + 0.158i)T + (-0.948 + 0.316i)T^{2} \)
41 \( 1 + (0.360 - 0.895i)T + (-0.721 - 0.692i)T^{2} \)
43 \( 1 + (0.316 - 0.948i)T^{2} \)
47 \( 1 + (0.568 - 0.822i)T^{2} \)
53 \( 1 + (-0.891 + 0.0539i)T + (0.992 - 0.120i)T^{2} \)
59 \( 1 + (-0.774 - 0.632i)T^{2} \)
61 \( 1 + (0.156 + 0.768i)T + (-0.919 + 0.391i)T^{2} \)
67 \( 1 + (-0.428 - 0.903i)T^{2} \)
71 \( 1 + (-0.960 + 0.278i)T^{2} \)
73 \( 1 + (-0.970 + 0.239i)T + (0.885 - 0.464i)T^{2} \)
79 \( 1 + (0.568 - 0.822i)T^{2} \)
83 \( 1 + (-0.970 + 0.239i)T^{2} \)
89 \( 1 + (1.90 + 0.509i)T + (0.866 + 0.5i)T^{2} \)
97 \( 1 + (-1.86 + 0.150i)T + (0.987 - 0.160i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.094321728034014629068316868204, −8.414953114670087278360369847492, −7.79979739575522082101644597621, −6.65235001989320212501479585378, −5.77409345483367184785842198790, −4.87661343649676251104509858134, −4.07162858047608357278861355897, −3.52435282406398842782156012558, −2.20502202655176383039217280552, −1.41293196513239512673111681398, 0.17826306496232969750028344064, 2.20667186933666542476828596296, 3.07930493642405207569326896929, 4.15171669528529638453925186622, 5.21497284189844712338161881060, 5.60508136041426465470250101052, 6.62348891595574539023181259284, 7.25043838135350491845888491198, 7.75964834361916291302906274283, 8.498803177426857718011466512879

Graph of the $Z$-function along the critical line